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A molecular orbital approach to chemisorption
SURFACE
SCIENCE 24 (1971) 191-208 0 North-Holland
Publishing Co.
A MOLECULAR ORBITAL APPROACH TO CHEMISORPTION I. ATOMLC HYDROGEN ON GRAPHITE ALAN J. BENNETT,
BRUCE
McCARROLL
and RICHARD
P. MESSMER
General Electric Research and Development Center, Schenectady, New York 12301, U.S.A. Received 19 June 1970 We examine the use of Extended Htickel Theory, a semi-empirical molecular orbital approach, in treating chemisorption. The calculations include all valence electrons and overlap integrals. Adsorbate energy levels are self consistently determined. The substrate is represented by a relatively small number of atoms which is shown to be adequate for obtaining semi-quantitative results. The small representation permits investigation of adsorbate interactions with both ideal and imperfect surfaces. The binding energy, binding sites, and barrier to surface mobility exhibited by atomic hydrogen on a graphite basal surface are obtained. The ex~rimentally observed formation of CH4 on adsorption of hydrogen is also considered. Calculations on electrophilic adsorbates reveal the necessity for a more completely self consistent treatment in which all energy levels are adjusted.
1. Introduction A variety of experiments on gas-solid systems indicate some of the desirable features required in a theory of chemisorption. The geometric arrangement of surface atoms is important as evidenced both by the multiple binding energies 1) exhibited by a particular adsorbate on a given single crystal surface of a particular substrate and by the variationsa) of those binding energies with the single crystal surface exposed to the gas phase. The results of surface mobility measurements3) also show this sensitivity of adsorption phenomena to the details of surface structure. Localized electronic configurations near the adsorbate atoms are also important. The electronic spectra*) obtained from ion neutralization measurements, for example, have a striking similarity to the atomic spectra of the adsorbate. Other experiments5), however, show that bulk electronic properties must also be included. Measurementsc) of the changes in work function with adsorbate coverage show that charges may be severely redistributed at the interface, and therefore a self-consistent determination of the potential and charge density is another vital ingredient in any satisfactory formulation of the properties of a substrate-adsorbate system. 191
192
ALAN
J. BENNETT
BRUCE
MCCARROLL
AND RICHARD
P. MESSMER
Theories of chemisorption can be conveniently classified as classical or quanta]. The degree to which the desirable features are included varies considerably. The classical approach usually assumes a two body interaction potential of the Morse or Lennard-Jones 12-6 form between the atoms that comprise the system7). The parameters that enter this potential are determined from gas phase or bulk properties of the relevant atoms. An advantage of this approach is that the simple force laws permit large numbers of substrate atoms to be included in the calculation. Although useful in rationalizing some experimental data, this approach fails to make contact with the quantum mechanical nature of real systems. In addition, there is evidences) to suggest that pair additive forces are often inadequate to rationalize the behavior of chemisorbed atoms. This casts further doubt on the future of such an approach. The various quantum treatments of chemisorption may be categorized by the wavefunctions chosen to represent the system. For example, a modified Heitler-London (valence bond) approximation has been used to examine a system consisting of H molecules adsorbed on carbons). The complexity of the calculation is such that usually two atoms represent the substrate: A model with more atoms requires very drastic approximation lo). Consequently, the geometric properties of the surface and the bulk properties of the solid substrate are very difficult to include in these calculationsll). A second approachi2-i5), that of molecular orbital theory, is particularly attractive in that its pictorial representation is a tempting means of predicting adsorption behavior16). Although rigorous treatment is again difficult, reasonably well defined and tractable approximations l7,l*) have been developed in the course of treating small molecules. Newnsra), and also Grimley15), have recently considered the binding energy of chemisorbed hydrogen. In their calculations, the electronic wavefunctions of the substrate (obtained in principle from a band structure calculation) interact with a hydrogen atomic s orbital via a tunneling matrix element19,aO). The substrate wavefunctions are found to affect the binding energy through a function A(E) which describes the width of the adsorbate level. This function can, in principle, be calculated and reflects the band structure and particular crystallographic orientation of the substrate. The various matrix elements of an effective Hamiltonian which enter this calculation are difficult to determine. Newns confines his actual calculations to a simple model in which A(E) is taken as that due to a single one dimensional tight binding band. Overlap integrals between substrate and adsorbate wavefunction are ignored in Newns’ work, although they may be quite large and important in determining cohesive energies 21). The present work is another application of molecular orbital theory to
A MOLECULAR
ORBITAL
APPROACH
TO CHEMISORPTION
193
chemisorption. We present the results of an examination of chemisorption with a semi-empirical quantum approach, the Extended Htickel Theory (EHT) developed extensively by Hoffmann22) and his collaborators, and a surface molecule concept introduced by Grimleyas). Although more sophisticated molecular orbital formulations exist, the EHT has been found reasonably adequate for comparing the binding energies of molecular isomers. We use the method here to study the relative binding energies of different adsorbate configurations. Both lattice and orbital geometry are naturally included in the treatment. Absolute values of the binding energies are not, however, very reliable. The results of these initial calculations serve both to demonstrate the usefulness of such a simplified theory and the necessity for various extensions and modifications of the method. Preliminary EHT calculations have recently been carried out for hydrogen adsorbed on lithium and nickel 24) and the nickel-hydrogen-ethylene systems25). The extended Hiickel approach is characterized by the use of: (i) a linear combination of atomic orbitals centered on both substrate and adsorbate atoms to represent the wavefunctions of the system, (ii) all non-zero overlap elements between various orbitals, and (iii) a particular prescription to calculate the matrix elements of the Hamiltonian. Hoffmann’s approach takes no account of charge redistributionss) and has most often been applied to organic molecules where little occurs. We introduce a simple means of including such effects which is consistent with the other approximations entering the theory. In the present work, only the adsorbate energy levels are self-consistently determined - an approximation adequate for hydrogen chemisorption. As discussed below, electrophilic adsorbates are found to require a more involved treatments7). A major difficulty in applying Hoffmann’s theory to the chemisorption system is the large number of atoms (and orbitals) in the substrate. Grimleyaa) has recently suggested an approach in which the interaction of the adsorbate and a limited number of substrate atoms is considered as a first approximation to the total chemisorption interaction. These results must then be corrected by considering, in a perturbative sense, the interaction of the surface molecule with the remainder of the substrate atoms. Recent calculations28) have indicated that the localized (surface) states predicted by molecular orbital theory using small finite models of one and two dimensional crystals are very much like those predicted by semi-in~nite models. In addition, the non-localized states predicted by such calculations are simply related to their infinite sample analoguessa). The use of the surface molecule concept in the context of extended Hiickel theory may therefore be expected to yield semi-quantitative results with a relatively small number of substrate atoms. Indeed, the comparison of results
194
ALAN
.I. BENNETT,
BRUCE
MCCARROLL
AND RICHARD
P. MESSMER
obtained with different size representations of the substrate serves to illustrate the adequacy of the surface molecule concept for semi-quantitative descriptions. The use of an adsorbate atom as a potential probe near the middle of the substrate representation approximately simulates the interaction of an atom with a perfect surface. At the boundary of the substrate representation, insight is obtained into the behavior of adatoms at edges and steps, and into the formation of compounds that involve removal of substrate atoms. The absence of explicit many body correlation effects in the EHT calculations precludes an accurate reproduction of the experimental work function. Then, too, the presence of surface states and the finite spacing of energy levels in a small representation affect its work function. The relative value of the substrate work function and the adsorbate energy levels is of crucial importance in determining the ionicity of their interaction. We shall choose here to assume that the small representation is a plateau region in electrical contact with the remainder of the substrate. Its energy levels must then be self-consistently adjusted so as to equalize its Fermi level with that of the bulk. A crude means of assuring this equality is introduced below. We consider, in this paper, the chemisorption of H on graphite substrates. Difficulties encountered in considering other adsorbates are also discussed. The graphite substrate was chosen for our first calculations since: (i) only s and p orbitals need be considered, (ii) the extended Hiickel theory has been most often used and tested on organic molecules rich in carbon, and (iii) a single two-dimensional sheet provides a reasonable representation of the substrateso-s2). Furthermore, simple atomic species do react with graphitessss4) in characteristic ways to yield simple molecular products (e.g. CH,, C,N,, CO and CO,). This may eventually permit detailed comparison between the results of calculation and experiment. The EHT computer program of Moore and Carlson 35) served as the starting point for our own programming efforts. After modification of that program, three minutes are required on a GE 635 processor to complete a calculation representing a 16 carbon lattice and hydrogens. Solution of a 32 carbon lattice problem requires 20 minutes of processor time. In the next section, we present a method of obtaining the extended Hiickel theory from the full Hamiltonian, carefully noting the various approximations required. Section 3.1 contains results for our representation of the substrate. Results for adsorbed hydrogen (section 3.2) and other species (section 3.3) are then presented. Methane formation from atomic hydrogen and graphite is discussed in section 3.4.
A MOLECULAR ORBITAL APPROACH
TO C~I~RPTION
195
2. Formalism
The extended Htickel theory was originally presented as a semi-empirical procedure. More recently attempts 36,37) have been made to rationalize it from consideration of the full Hartree-Fock Hamiltonianss) for closed shell systems. We confine our attention here to such systems. Although the approach will be used in other situations, the nature of the approximations involved is revealed by the present discussion. The wavefunctions are antisymmetrized products of one electron orbitals ‘pi which are taken as
where the x1 are real atomic orbitals centered on the atoms of the system. The coefficients cil and eigenvalues Ei are then determined by solving
(2.2)
llr;,, - E&A = 0, where S is the overlap matrix associated with the (xn]
(2.3)
S&Y= 0 I 0) ’ The elements of Fare specified by
where pfiy = 2
2
ctciy
I
(2.5)
Y
occupied
=
s
x~(r~)x.(~,)~~x~(~~)x,(~~)di;dV2.
w5)
2, and R, denote the core charge and position respectively of the ccth atom. Solution of eq. (2.2) makes possible a calculation of: (i) the occupation of the ,&h atomic orbital: P,,@; (ii) the number of electrons on a given atom A: NA
=
1 p,, p afom A
+
c p~~s~v v+r
;
(iii) the reduced overlap populationQQ) between atoms A and B: QAB G
2
C
PpYSsY -
The latter quantity is a measure of the covalent bonding between the atoms. We consider atoms with one relevant principal quantum number. We first examine the diagonal terms of F and apply the Mulliken approxi-
196
mations40)
ALAN
J. BENNETT,
BRUCE
to the coulomb
For L centered
MCCARROLL
integral
AND RICHARD
P. MESSMER
and its associated
exchange
integral.
on atom B
where (2.9) The last three terms contain two center integrals. Conventional those terms and assumes that the Pp,, associated with atom atomic values. Koopman’s theorem then implies that
EHT ignores B have their
where 1, is the valence state ionization potentialhi) of an electron in orbital L on atom A, i.e. the ionization potential of the atom in the state characteristic of its molecular environment. In chemisorption systems, charge transfer can at times be large and must be considered. Charge transferred to (from) the adsorbate atom is taken from (placed into) metallic electron states which are not well localized. Consequently, possible large charge deviations are expected only on the adsorbate orbitals. If Pi:” is the normal occupation of the pth orbital of the valence state atom, we obtain (2.10) For the case of hydrogen, FALz - 13.6 + Pnlynni. We now consider
the off-diagonal
matrix elements
(2.11) of the Hamiltonian.
The
use of eqs. (2.7) yields
F,, g (4 -
JV2-
zA Ir
_
Ia>
RAI
+
c
Pp,Y,Ac7 +
+
c
pwY,l~
atoms A, B
atoms A, B
p~“spvYpla -
lr#v
c
P+
P
c&I
a#A
Ia>,
(2-W
A MOLECULAR
with 1~) centered
F&l = (4 -
ORBITAL
APPROACH
197
TO CHEMISORPTION
on atom A, and
ZB
9’ -
,r _
IA>+
RBj
c
c
P,Jwl
P,,Y,d +
P
atoms A, B
afomsA,B
Pp&YpoA - (4
+
9
c
(2.13)
c a#B
P#V with 12) centered on atom B. Noting that for real orbitals
(2.14)
+ F,,) 3
F,, = * P,,
again using eqs. (2.7) and assuming that x0 and xi are eigenfunctions resulting effective one electron Hamiltonian, we obtain L&[-r,+
c
(P,,-P;::mB)Yrll-I,+
c
P atomB
(P,,-P;:o~*)Y,..]s,O
P
atom A Z, Ir--R,I
(4 -4
of the
Ia>
+
:
c a=A,B
P atom A
PfipSlaYpaa+
++ c
P atom B
c P+
P,,Yh
atom A, B
PlrVSPVY/Ll~ +
+:
p,,skJ,n, c
c
01 m
Z,
Ia>.
(2.15)
c a#A,B
P#V
The last five terms contain three center integrals or the product of two center integrals and S. They are ignored in conventional EHT. The first term is explicitly included. The difference (P,, - PiFm) is, however, set equal to zero. The second term is approximated by multiplying the first by a constant K. We follow that general procedure. The difference between the atomic and molecular occupations is, however, maintained in our formulae. The conventional valuez2) of K= 1.75 - in very rough accord with viral theorem arguments - is used. The various results are not critically affected by small changes in that parameter. The binding energy of a molecular system is obtained by comparing its total energy with that of the isolated constituent atoms. A schematic HartreeFock expression for that binding energy is E,=
xEy’-(i
c Ey-) j
- (V$” - v$‘,)
+ v,, ,
(2.16)
198
ALAN
J. BENNETT,
BRUCE
MCCARROLL
AND RICHARD
P. MESSMER
where E, are the one electron energies computed approximately in eq. (2.2), V,, is the electron-electron repulsive energy, and V,, is the nuclear-nuclear repulsive energy. The seond term in eq. (2.16) corrects for the overcounting of the electron-electron interaction by the first. If the second and third term cancel, the binding energy may clearly be obtained by a simple sum over orbital energies. Goodisman42) has shown that such a cancellation approximately occurs and is a consequence of the isoelectronic principle. A slow variation of the approximate cancellation with internuclear separation would justify the neglect of those terms in calculations of relative binding energies. The binding energies reported here are determined from just the first term in (2.16). 3. Results and discussion 3.1. GRAPHITE SUBSTRATE All EHT calculations have used Slater orbitals to represent the atomic orbitals in eq. (2.1):
where parameters 2, and M, are taken from atomic data as are the ionization potentials which enter eq. (2.10). in an attempt to include solid state effects, parameters were initially obtained from a comparison with a tight-binding calculationss) of the twodimensional band structure of bulk graphite. Due to the weak coupling between hexagonal layers, it has been found that two dimensional calculations are reasonably adequate to account for many of the bulk properties of graphite with the notable exception of the conductivity, Bassani and Parraviciniss) obtained a two dimensional tight-binding band structure by fitting
I
2
Fig. 1. Sixteen carbon (16C) representation of a graphite lattice showing electron charge and reduced overlap populations (italics); underljned charge densities are those obtained with a hydrogen atom located 1 8, above position A.
A MOLECULAR
ORBITAL
a set of Sij and Hij to bulk optical
APPROACH
TO CHEMISORPTION
199
data. The use of values for the Z,, n,,
and I, derived from this parameterization lead to very diasappointing sults: In particular, the lattice is found not to be stable to deformation.
reThis
behavior emphasized that a parameterization designed to reproduce one property may be unsuitable for the proper simulation of another property42). As seen below, the customary use of the atomic parameters 22, 43) Z, = Z, = = 1.625, n = 3, I, = 19.44 eV, and IP = 10.67 eV, for the single s and three p orbitals chosen to represent the carbon atoms is more successful. Calculations were performed on various representations of the (0001) graphite substrate. The single layer 16 carbon representation of that substrate (16C) shown in fig. 1 was used in most of our work. Some results, however, were compared with those obtained using the single layer 32 carbon (32C) and stacked 32 carbon (32CS) representations shown in figs. 3 and 2 respectively. In table la, the binding energy/atom obtained using the different substrate models is listed. The agreement with the experimental value of -5 eV44) is reasonably good and the various results are similar. The 16C representation was used to calculate the binding energy/atom as a function of a, the nearest
Fig. 2. Stacked thirty two carbon (32CS) representation of a graphite lattice. Reduced overlap populations and charge distributions are the same, to three significant figures, as those shown in fig. 1.
Fig. 3.
Thirty two carbon (32C) representation of a graphite lattice showing electron charge and reduced overlap populations (italics).
200
ALAN .I. BENNETT, BRUCE MCCARROLL AND RICHARD P. MESSMER TABLE 1 (a) Representation
Binding
energy/atom
16C
4.17
32C 32CS
4.32 4.17
(eV)
(b) a (4 1.13
Binding
energy/atom
1.28
2.88 3.94
1.42
4.17
1.56
3.90 3.42
1.70
(eV)
neighbor distance. The results shown in table 1b indicate that the lattice is stable for a - 1.4 A which is the experimentally observed separation. The reduced overlap populations between various carbon atoms are shown along the lines connecting the atoms in figs 1 and 3. The reduced overlap population between interior atoms is - 1.O. A comparison with the calculated values for hydrocarbons indicates that this represents a bond order of - 1.5 between those atoms. A bond order of 1.33 is expected for the graphite structure45). Coulson 46) has considered edge rearrangement in two dimensional graphite lattices, and hypothesized that atoms such as those labeled 1 and 2 are shifted (as shown by dotted lines in fig. 1) in order to increase the lattice stability by creation of an additional bond. Our results suggest that such rearrangements are energetically unfavorable. An examination of the eigenenergies and wave functions obtained from eq. (2.2) indicates that the states may be divided into two classes. The band states extend through the substrate representation and have energies which may be correlated with the bulk band structure. The surface states47) are confined to the perimeter of the representation and have energies either within or without the range of band state energies. Their number may be roughly correlated with the number of dangling bonds on the perimeter. For large representations, these surface states would constitute a small fraction of the total number of states to be filled. They would then have a negligible effect on the Fermi level and the charge distribution in the region away from the perimeter. For the substrate representations used here, their effect is large. The range and distribution of band eigenvalues calculated differ from those
A MOLECULAR
given by Bassani adjustments structure.
ORBITAL
and Parravicinisa).
permitted,
precludes
APPROACH
201
TO CHEMISORPTION
Our use of atomic parameters, an accurate
reproduction
with no
of their
band
The total pi-electron
band width composed of p orbitals directed out of the plane is found to be N 11.61 eV (12.3 eV for 32C representation) N 17.8 eV (17.0 eV for the 32C representation) and the total filled bandwidth as compared to Bassani and Parravicini’s values of 13.6 eV and 14.3 eV respectively. The highest filled energy level occurs at - 10.0 eV (- 10.4 eV for the 32C representation) as opposed to the experimental value of N -4.8 eV48). This rather severe lack of agreement is not surprising, since many body correlation effects are not explicitly included in our calculations. For the small representations used here two other effects change the work function. When obvious surface (i.e. perimeter) states are ignored in filling the energy levels, the highest occupied level is moved toward the experimental value. These surface states are responsible for the enhanced electron charge on perimeter atoms shown in figs. 1 and 3. Also the finite spacing of the energy levels in a small representation affects the calculated work function. If the finite representation is assumed to be in electrical contact with the remainder of the substrate, thermodynamics demands that its Fermi level and that of the substrate be equal. Consequently electrons flow from the bulk into the surface region to create a charge excess which causes a Hartree potential. That potential self-consistently adjusts the representation’s energy levels in order to control the charge flow. Simple Poisson equation considerations indicate that a small excess of charge (6 1 cl/atom) is sufficient to shift the energy levels on the order of electron volts. 3.2. HYDROGEN CHEMISORPTION The diagonal matrix element associated with the hydrogen s orbital is given by eq. (2.4). The magnitude of yiil can be calculated using hydrogenic orbitals and found to be 8.5 eV. As suggested by Newns, we use ylnn = + (I” -AH) = 6.45 eV, a somewhat better measure of the physically desired quantity since it takes some account of correlation effects. For hydrogen positions near a metal surface, even greater reductions in yiln may occur49). We consider the interaction of the adsorbate with the substrate representation in electrical contact with the bulk, i.e. with a plateau region on an otherwise perfect surface. We demand that the highest filled energy level, EFs, in the substrate representation be close to the correct bulk Fermi level EFB = - 4.8 eV. The simplest means of assuring this equality is a shift of each representation level by (E,, -I$,). A severe and probably artificial reduction in the cohesive energy of the substrate representation would then be obtained from eq. (2.2) et seq. by use of the modified carbon levels.
202
ALAN
.I. BENNETT,
BRUCE
MCCARROLL
AND
RICHARD
The physically correct difference between adsorbate crucial for investigating the ionicity of their interaction,
I’. MESSMER
and substrate levels, can alternatively be
maintained by changing the effective adsorbate levels by - (E,, -I&). This approach is used here. There are resultant changes in the magnitude of various calculated binding energies, but the relative binding energies which are the main output of our calculations remain approximately invariant. For investigation of perfect surfaces, an approach, currently under investigation, is to assume periodic boundary conditions in the plane, and thus eliminate surface states. The self-consistent charge and binding energies were found after two or three iterations. The hydrogen binding energies as a function of the distance Z above various points in the lattice plane are shown in figs. 4a, 4b, and 4c for the point A (at a carbon atom), point B (between two nearest neighbor carbon atoms), and point C (in the middle of a hexagonal cell). The points A, B, and C are indicated in fig. 1. The dashed lines in fig. 4 indicate the charge on the hydrogen adsorbate. The calculations were performed on the 16C representation but some results for the 32C representation are shown by crosses in the figure. (0)
‘H -c
5
uY
__---
0.5
1.0
;i
I% E B 1.5 g
-I
-I t
ZCil Fig. 4. The solid lines show binding energy versus distance from the graphite surface for atomic hydrogen located above points (a) A, (b) B, and (c) C on the 16C representation. The dashed lines in the figure indicate the net hydrogen charge. The crosses show the results of calculations on the 32C representation.
A MOLECULAR
The
converged
effective
ORBITAL
APPROACH
hydrogen
203
TO CHEMISORPTION
ionization
potentials
are rather
close
to the atomic value of 13.6 eV. The conventional, and generally successful, EHT use of that value would thus not strongly affect our results. The calculation at the point A is clearly inadequate for very small values of 2 when the adsorbate and carbon atom coincide. At the other extreme, molecular orbital calculations without configuration interaction fail to give the correct dissociation behavior at large separations. The underlined numbers in fig. 1 show the charge distribution on the 16C lattice when a hydrogen atom is located 1 A above the point A. Comparison with the other charges shown in that figure indicates that little charge redistribution occurs due to the presence of the hydrogen. This justifies the self-consistent adjustment of only the hydrogen atomic energy level. Fig. 4 and similar curves for other points indicates that the maximum binding always occurs for Z-1 A. An equi-binding energy surface obtained using the 16C representation is shown in fig. 5. We again emphasize that differences between the binding energies shown in the figure are of most significance. The maximum binding occurs above a carbon atom. The barrier for hydrogen motion along the lines connecting nearest neighbor carbons is low, and hydrogen should avoid the center of the hexagonal cell in preference for positions above the carbon atoms. The use of larger representations is found to affect some details but not the essential features of this figure.
C
C
c 1.121.09
I.14 Fig. 5.
1.05
1.09 I.12
c I.14
Equi-binding energy contours (eV) for atomic hydrogen adsorbed on the (16C) graphite substrate. Positive energies correspond to binding.
204
ALAN
This prediction
J. BENNETT,
contrasts
BRUCE
MCCARROLL
AND RICHARD
sharply with conventional
P. MESSMER
hard sphere reasoning,
which maintains that adsorbed species tend to assume a surface configuration that maximizes the number of nearest substrate atoms. Our result suggests that orbital geometry is a crucial ingredient in the determination of surface binding sites. 3.3. OTHERADSORBATES Calculations were performed for nitrogen and oxygen adsorbates similar to those performed for hydrogen adsorbates. However, the presence of such electrophilic species has a profound effect on the charge distribution of the substrate representation. Self-consistent adjustment of only the adsorbate matrix elements was found inadequate to obtain convergence. Self-consistent adjustment of substrate atom matrix elements is apparently required. The results of preliminary calculations do indicate that electrophilic adsorbate atoms, in contrast to hydrogen, are not repelled at the center of the hexagonal graphite cell. Preliminary calculations were also carried out for other species. For example, the results predict that CO is not adsorbed on top of the substrate but seems to bind only at edge atoms with the carbon atom of the CO molecule bonding to a carbon atom of the substrate. LEED results 50) suggest that CO does not bind on top, and other experimental data indicate that edgebound CO may be a participating species in the oxidation of carbon51). Despite this encouraging comparison between theory and experiment, it must be remembered that no self-consistent adjustments were attempted in these calculations, so that the conclusions for the CO-graphite system must be taken as tentative. 3.4. METHANE
FORMATION
The results of experimental studies on the interaction of hydrogen atoms with graphite and carbon powders indicate that the main reaction product is gaseous methane CH,s*), which is in accord with thermodynamic expectations. A quantum mechanical examination of a reaction path should predict intermediate steps between the starting materials and the final reaction products. At each step, the geometric configuration of maximum stability should be apparent, and any energetic barriers to progress through the sequence of steps should emerge from the calculations. In principle, all viable alternate reaction paths should be calculated. Of course, the energetics of the terminal configurations (products) must be consistent with thermodynamic predictions. In practice, the possible geometric alternatives of a given step of even our
A MOLECULAR
simple system is enormous,
ORBITAL
APPROACH
TO CHEMISORPTION
and the detailed investigation
205
of all configurations
is clearly impractical. Therefore, the reaction path outlined below for the formation of CH, from hydrogen and a graphite crystal must be considered illustrative rather than definitive. The configurations considered are those of lowest energy picked from a number of reasonable possible alternatives. A single hydrogen atom binds with an energy of about 4 eV to edge carbon atoms, i.e. much more strongly than to atoms in the center of the representation. This behavior suggests that the strongest interactions occur at the edges and no doubt other such portions of a real crystal surface where so-called free valences are available. Indeed, reactions of atomic hydrogen with crystalline graphite surfaces produce surface pits whose size increases with increasing exposure to hydrogen atomss2). The experimental behavior thus also suggests that the abstraction of carbon atoms from the surface occurs initially at surface imperfections, and then continues at edges thus formed. The stabilities shown in fig. 6 are determined by subtracting from the calculated energy of a given configuration the sum of the total energy for a bare 16 carbon lattice and - 1.36 n where II is the number of bound hydrogens. The 16C lattice and n-gas phase hydrogens is therefore the zero of energy for the comparisons. Only rough charge consistency was obtained.
*
METHANE
Fig. 6.
Simulated reaction path for the formation of methane from a 16C lattice and 8 hydrogen atoms; see text.
206
ALAN
J. BENNETT,
BRUCE
MCCARROLL
AND RICHARD
P. MESSMER
The addition of the first hydrogen is most stable on carbon D (fig. 6). It lies in the plane of the sheet. The second hydrogen binding at E provides the most stability and the addition of the third hydrogen provides the planar arrangement shown in the figure. The addition of more hydrogens with the aim of incorporating carbon D into a methane molecule proceeds as shown in the figure. Any carbon with two hydrogens has them equidistant above and below the plane of the sheet with an H-C-H angle of 120 O.The three hydrogens on carbon D are situated at three corners of a tetrahedron with carbon D as the center. The most stable aspect is with one of the three hydrogens in the plane of the sheet. The fourth hydrogen is added to carbon D from above the plane. Several classes of configurations were not examined. In particular, as hydrogens are added to the system, it is reasonable to expect the C-C bond length to increase to approximately 1.54 A, which would parallel behavior observed for organic molecules of increasing hydrogenation. However, other than moving carbon D away from the lattice (fig. 6) in the plane, no attempt is made to adjust C-C bond lengths as the hydrogen adsorption progresses. Out-of-plane distortions in carbon rings that may occur upon increasing hydrogenation are also ignored. The events depicted certainly constitute a reaction path. But they undoubtedly do not constitute the most probable reaction path, except fortuitously. However, the extended Hiickel calculations do enable us to energetically distinguish between configurations for a fixed number of added hydrogens and exhibit at least one energetically favorable means of methane production. Further, the results of calculations for terminal configurations involving CH, or C,H, as the products reflect both the thermodynamic predictions and the experimental results of methane as the most stable product. 4. Summary An extended Hiickel theory has been used to treat chemisorption. The present work is characterized by the self-consistent adjustment of only adsorbate atom energy levels and the use of a finite representation of the substrate. The former approximation seems adequate for treating atomic hydrogen on graphite. However, electrophilic adsorbates require a more complete self-consistent treatment in which all energy levels are adjusted. The finite representation of the substrate does allow semiquantitative investigations of adsorption on both perfect and imperfect surfaces. For small representations the population of electron states localized near the perimeter can significantly perturb the distribution of electrons from that of a larger representation. We emphasize that absolute values of the adsorbate binding
A MOLECULAR
ORBITAL
APPROACH
TO CHEMISORPTION
207
energies are only crudely determined. The calculated differences between the energies of various configurations are, however, relatively invariant to various details of the procedure. This is of significance because many useful predictions can be made on the basis of relative binding energies. For example, a feature that clearly emerges from the results of our calculations is that the use of lattice geometry alone is insufficient to predict the binding site on a given surface for a given adsorbate. The directional characteristics of the electronic interactions are indeed controlling, so that, for example, a position on the surface that maximizes the number of nearest neighbors is not always a position of minimum energy. Although fraught with technical problems the simple EHT method yields results which suggest that with suitable modifications of the method we may eventually hope for detailed predictions of various adsorbate properties.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11)
12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25)
See, for example, T. A. DeIchar and G. Ehrlich, J. Chem. Phys. 42 (1965) 2686. E. W. Plummer and T. N. Rhodin, J. Chem. Phys. 49 (1968) 3479. G. Ehrhch and F. Hudda, J. Chem. Phys. 44 (1966) 1039. H. D. Hagstrum and G. E. Becker, Phys. Rev. Letters 22 (1969) 1054. G. C. Bond, Cutulyis &y Merals (Academic Press, New York, 1962). E.g. L. D. Schmidt and R. Gomer, J. Chem. Phys. 45 (1966) 1605. E.g. H. E. Neustadter and R. J. Bacigalupi, Surface Sci. 6 (1967) 246. G. Ehrlich, H. Heyne and C. F. Kirk, in: The Sfructare and Chemistry ofSolid Sarfaces, Ed. G. A. Somorjai (Wiley, New York, 1969) p. 49-l. S. Glasstone, K. J. Laidler and H. Eyring, The Theory of Rate Processes (McGraw Hill, New York, 1941). J. Horiuti and K. Miiller, J. Res. Inst. Catalysis, Hokkaido Univ. 16(1968) 605. R. Gomer and J. R. Schrieffer (private communication) have recently formulated a many electron Heitler-London approach which is being applied to the chemisorption problem. T. Toya, J. Res. Inst. Catalysis, Hokkaido Univ. 8 (1960) 209; 6 (1958) 308. K. F. Wojeiechowski, Acta Phys. Polonica 29 (1966) i19; 33 (1968) 363. D. M. Newns, Phys. Rev. 178 (1969) 1123. T. B. Grimley, Proc. Phys. Sot. (London) 90 (1967) 751; 92 (1967) 776. D. A. Dowden, in: Coloquio Sobre Quimica Fisica de Procesos en Superficies Solidas (Libreria Cientiflca Medinaceli, Madrid, 1965). K. Jug, Theoret. Chim. Acta. (Berlin) 14 (1969) 91. H. H. Jaffe, Accounts of Chemical Research 2 (1969) 136. A. J. Bennett and L. M. Falicov, Phys. Rev. 151 (1966) 512. J. W. Gadzuk, in: The Structure and C~e~jstry of Solid Surfaces, Ed. G. A. Somorjai (Wiley, New York, 1969) p. 43-l. P. 0. Lowdin, Advan. Phys. 5 (1956) 1. R. Hoffmann, J. Chem. Phys. 36 (1962) 2179; 36 (1962) 2189; 40 (1963) 2474. T. B. Grimley, in: Molecular Processes on Solid Surfaces, Eds. E. Drauglis, R. D. Gretz and R. J. Jaffee (McGraw-Hill, New York, 1969) p. 299. P. Markov and D. Lazarov, Compt. Rend. Acad. Butgare Sci. 22 (1969) 455; D. Shopov, A. Andreev and D. Petkov, J. Catalysis 13 (1969) 123. A. Rosenthal and R. Hoffmann (private communication).
208
ALAN
J. BENNETT,
BRUCE
MCCARROLL
AND RICHARD
P. MESSMER
26) B. J. Duke, Theoret. Chim. Acta (Berlin) 9 (1968) 260. 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 3% 39) 40)
41) 42) 43) 44) 45) 46) 47) 48) 49) 50) 51) 52)
A. J. Bennett, B. McCarroll and R. P. Messmer, to be published. G. Blyholder and C. A. Coulson, Trans. Faraday Sot. 63 (1967) 1782. B. R. Cooper and A. J. Bennett, Phys. Rev., to be published. J. C. Slonczewski and P. R. Weiss, Phys. Rev. 109 (1958) 272. J. W. McClure, Phys. Rev. 108 (1957) 612. F. Bassani and G. Parravicini, Nuovo Cimento 50B (1967) 95. F. J. Vastola, P. L. Walker and J. P. Wightman, Carbon 1 (1963) 11. P. S. Gill, R. E. Toomey and H. C. Moser, Carbon 5 (1967) 43. E. B. Moore Jr. and C. M. Carlson, in: Lattice Defects and Their Inleructions, Ed. R. R. Hasiguti (Gordon and Breach, New York, 1967) p. 167. G. Blyholder and C. A. Coulson, Theoret. Chim. Acta (Berlin) 10 (1968) 316. M. D. Newton and W. N. Lipscomb, J. Am. Chem. Sot. 88 (1966) 2353, 2361, 2367. C. C. J. Roothaan, Rev. Mod. Phys. 23 (1951) 69. R. S. Mulliken, J. Chem. Phys. 23 (1955) 1833. R. S. Mulliken, J. Chim. Phys. 46 (1949) 497, 675.; J. W. Richardson and R. E. Rundle, Report ISC-830, Ames Lab., Iowa State College USAEC, Technical Information Extension, Oak Ridge, Tenn. 1956. R. S. Mulliken, J. Chem. Phys. 2 (1934) 782. J. Goodisman, J. Am. Chem. Sot. 91 (1969) 6552. J. A. Pople and G. A. Segal, J. Chem. Phys. 43 (1965) 5136. M. A. Kanter, Phys. Rev. 107 (1957) 655. L. Pauling, The Nature ofthe Chemical Bond (Cornell University Press, Ithaca, 1960). C. A. Coulson, in: Proc. Fourth Conf. on Carbon (Pergamon Press, New York, 1960) p. 215. J. Koutecky and M. Tomasek, Phys. Rev. 120 (1960) 1212. V. S. Fomenko, Handbook of Thermionic Properties (Plenum Press Data Division, New York, 1966). A. Heeger, in: Solid Rate Physics, Vol. 23 (Academic Press, New York, 1969). J. J. Lander and J. Morrison, Surface Sci. 6 (1967) 1. See, for example, L. Schmidt, H. P. Boehm and U. Hoffman, in: Proc. Third Conf. on Carbon (Pergamon Press, New York, 1959) p. 235. B. McCarroll and D. W. McKee, Nature 225 (1970) 723; Carbon, to be published.
SCIENCE 24 (1971) 191-208 0 North-Holland
Publishing Co.
A MOLECULAR ORBITAL APPROACH TO CHEMISORPTION I. ATOMLC HYDROGEN ON GRAPHITE ALAN J. BENNETT,
BRUCE
McCARROLL
and RICHARD
P. MESSMER
General Electric Research and Development Center, Schenectady, New York 12301, U.S.A. Received 19 June 1970 We examine the use of Extended Htickel Theory, a semi-empirical molecular orbital approach, in treating chemisorption. The calculations include all valence electrons and overlap integrals. Adsorbate energy levels are self consistently determined. The substrate is represented by a relatively small number of atoms which is shown to be adequate for obtaining semi-quantitative results. The small representation permits investigation of adsorbate interactions with both ideal and imperfect surfaces. The binding energy, binding sites, and barrier to surface mobility exhibited by atomic hydrogen on a graphite basal surface are obtained. The ex~rimentally observed formation of CH4 on adsorption of hydrogen is also considered. Calculations on electrophilic adsorbates reveal the necessity for a more completely self consistent treatment in which all energy levels are adjusted.
1. Introduction A variety of experiments on gas-solid systems indicate some of the desirable features required in a theory of chemisorption. The geometric arrangement of surface atoms is important as evidenced both by the multiple binding energies 1) exhibited by a particular adsorbate on a given single crystal surface of a particular substrate and by the variationsa) of those binding energies with the single crystal surface exposed to the gas phase. The results of surface mobility measurements3) also show this sensitivity of adsorption phenomena to the details of surface structure. Localized electronic configurations near the adsorbate atoms are also important. The electronic spectra*) obtained from ion neutralization measurements, for example, have a striking similarity to the atomic spectra of the adsorbate. Other experiments5), however, show that bulk electronic properties must also be included. Measurementsc) of the changes in work function with adsorbate coverage show that charges may be severely redistributed at the interface, and therefore a self-consistent determination of the potential and charge density is another vital ingredient in any satisfactory formulation of the properties of a substrate-adsorbate system. 191
192
ALAN
J. BENNETT
BRUCE
MCCARROLL
AND RICHARD
P. MESSMER
Theories of chemisorption can be conveniently classified as classical or quanta]. The degree to which the desirable features are included varies considerably. The classical approach usually assumes a two body interaction potential of the Morse or Lennard-Jones 12-6 form between the atoms that comprise the system7). The parameters that enter this potential are determined from gas phase or bulk properties of the relevant atoms. An advantage of this approach is that the simple force laws permit large numbers of substrate atoms to be included in the calculation. Although useful in rationalizing some experimental data, this approach fails to make contact with the quantum mechanical nature of real systems. In addition, there is evidences) to suggest that pair additive forces are often inadequate to rationalize the behavior of chemisorbed atoms. This casts further doubt on the future of such an approach. The various quantum treatments of chemisorption may be categorized by the wavefunctions chosen to represent the system. For example, a modified Heitler-London (valence bond) approximation has been used to examine a system consisting of H molecules adsorbed on carbons). The complexity of the calculation is such that usually two atoms represent the substrate: A model with more atoms requires very drastic approximation lo). Consequently, the geometric properties of the surface and the bulk properties of the solid substrate are very difficult to include in these calculationsll). A second approachi2-i5), that of molecular orbital theory, is particularly attractive in that its pictorial representation is a tempting means of predicting adsorption behavior16). Although rigorous treatment is again difficult, reasonably well defined and tractable approximations l7,l*) have been developed in the course of treating small molecules. Newnsra), and also Grimley15), have recently considered the binding energy of chemisorbed hydrogen. In their calculations, the electronic wavefunctions of the substrate (obtained in principle from a band structure calculation) interact with a hydrogen atomic s orbital via a tunneling matrix element19,aO). The substrate wavefunctions are found to affect the binding energy through a function A(E) which describes the width of the adsorbate level. This function can, in principle, be calculated and reflects the band structure and particular crystallographic orientation of the substrate. The various matrix elements of an effective Hamiltonian which enter this calculation are difficult to determine. Newns confines his actual calculations to a simple model in which A(E) is taken as that due to a single one dimensional tight binding band. Overlap integrals between substrate and adsorbate wavefunction are ignored in Newns’ work, although they may be quite large and important in determining cohesive energies 21). The present work is another application of molecular orbital theory to
A MOLECULAR
ORBITAL
APPROACH
TO CHEMISORPTION
193
chemisorption. We present the results of an examination of chemisorption with a semi-empirical quantum approach, the Extended Htickel Theory (EHT) developed extensively by Hoffmann22) and his collaborators, and a surface molecule concept introduced by Grimleyas). Although more sophisticated molecular orbital formulations exist, the EHT has been found reasonably adequate for comparing the binding energies of molecular isomers. We use the method here to study the relative binding energies of different adsorbate configurations. Both lattice and orbital geometry are naturally included in the treatment. Absolute values of the binding energies are not, however, very reliable. The results of these initial calculations serve both to demonstrate the usefulness of such a simplified theory and the necessity for various extensions and modifications of the method. Preliminary EHT calculations have recently been carried out for hydrogen adsorbed on lithium and nickel 24) and the nickel-hydrogen-ethylene systems25). The extended Hiickel approach is characterized by the use of: (i) a linear combination of atomic orbitals centered on both substrate and adsorbate atoms to represent the wavefunctions of the system, (ii) all non-zero overlap elements between various orbitals, and (iii) a particular prescription to calculate the matrix elements of the Hamiltonian. Hoffmann’s approach takes no account of charge redistributionss) and has most often been applied to organic molecules where little occurs. We introduce a simple means of including such effects which is consistent with the other approximations entering the theory. In the present work, only the adsorbate energy levels are self-consistently determined - an approximation adequate for hydrogen chemisorption. As discussed below, electrophilic adsorbates are found to require a more involved treatments7). A major difficulty in applying Hoffmann’s theory to the chemisorption system is the large number of atoms (and orbitals) in the substrate. Grimleyaa) has recently suggested an approach in which the interaction of the adsorbate and a limited number of substrate atoms is considered as a first approximation to the total chemisorption interaction. These results must then be corrected by considering, in a perturbative sense, the interaction of the surface molecule with the remainder of the substrate atoms. Recent calculations28) have indicated that the localized (surface) states predicted by molecular orbital theory using small finite models of one and two dimensional crystals are very much like those predicted by semi-in~nite models. In addition, the non-localized states predicted by such calculations are simply related to their infinite sample analoguessa). The use of the surface molecule concept in the context of extended Hiickel theory may therefore be expected to yield semi-quantitative results with a relatively small number of substrate atoms. Indeed, the comparison of results
194
ALAN
.I. BENNETT,
BRUCE
MCCARROLL
AND RICHARD
P. MESSMER
obtained with different size representations of the substrate serves to illustrate the adequacy of the surface molecule concept for semi-quantitative descriptions. The use of an adsorbate atom as a potential probe near the middle of the substrate representation approximately simulates the interaction of an atom with a perfect surface. At the boundary of the substrate representation, insight is obtained into the behavior of adatoms at edges and steps, and into the formation of compounds that involve removal of substrate atoms. The absence of explicit many body correlation effects in the EHT calculations precludes an accurate reproduction of the experimental work function. Then, too, the presence of surface states and the finite spacing of energy levels in a small representation affect its work function. The relative value of the substrate work function and the adsorbate energy levels is of crucial importance in determining the ionicity of their interaction. We shall choose here to assume that the small representation is a plateau region in electrical contact with the remainder of the substrate. Its energy levels must then be self-consistently adjusted so as to equalize its Fermi level with that of the bulk. A crude means of assuring this equality is introduced below. We consider, in this paper, the chemisorption of H on graphite substrates. Difficulties encountered in considering other adsorbates are also discussed. The graphite substrate was chosen for our first calculations since: (i) only s and p orbitals need be considered, (ii) the extended Hiickel theory has been most often used and tested on organic molecules rich in carbon, and (iii) a single two-dimensional sheet provides a reasonable representation of the substrateso-s2). Furthermore, simple atomic species do react with graphitessss4) in characteristic ways to yield simple molecular products (e.g. CH,, C,N,, CO and CO,). This may eventually permit detailed comparison between the results of calculation and experiment. The EHT computer program of Moore and Carlson 35) served as the starting point for our own programming efforts. After modification of that program, three minutes are required on a GE 635 processor to complete a calculation representing a 16 carbon lattice and hydrogens. Solution of a 32 carbon lattice problem requires 20 minutes of processor time. In the next section, we present a method of obtaining the extended Hiickel theory from the full Hamiltonian, carefully noting the various approximations required. Section 3.1 contains results for our representation of the substrate. Results for adsorbed hydrogen (section 3.2) and other species (section 3.3) are then presented. Methane formation from atomic hydrogen and graphite is discussed in section 3.4.
A MOLECULAR ORBITAL APPROACH
TO C~I~RPTION
195
2. Formalism
The extended Htickel theory was originally presented as a semi-empirical procedure. More recently attempts 36,37) have been made to rationalize it from consideration of the full Hartree-Fock Hamiltonianss) for closed shell systems. We confine our attention here to such systems. Although the approach will be used in other situations, the nature of the approximations involved is revealed by the present discussion. The wavefunctions are antisymmetrized products of one electron orbitals ‘pi which are taken as
where the x1 are real atomic orbitals centered on the atoms of the system. The coefficients cil and eigenvalues Ei are then determined by solving
(2.2)
llr;,, - E&A = 0, where S is the overlap matrix associated with the (xn]
(2.3)
S&Y= 0 I 0) ’ The elements of Fare specified by
where pfiy = 2
2
ctciy
I
(2.5)
Y
occupied
=
s
x~(r~)x.(~,)~~x~(~~)x,(~~)di;dV2.
w5)
2, and R, denote the core charge and position respectively of the ccth atom. Solution of eq. (2.2) makes possible a calculation of: (i) the occupation of the ,&h atomic orbital: P,,@; (ii) the number of electrons on a given atom A: NA
=
1 p,, p afom A
+
c p~~s~v v+r
;
(iii) the reduced overlap populationQQ) between atoms A and B: QAB G
2
C
PpYSsY -
The latter quantity is a measure of the covalent bonding between the atoms. We consider atoms with one relevant principal quantum number. We first examine the diagonal terms of F and apply the Mulliken approxi-
196
mations40)
ALAN
J. BENNETT,
BRUCE
to the coulomb
For L centered
MCCARROLL
integral
AND RICHARD
P. MESSMER
and its associated
exchange
integral.
on atom B
where (2.9) The last three terms contain two center integrals. Conventional those terms and assumes that the Pp,, associated with atom atomic values. Koopman’s theorem then implies that
EHT ignores B have their
where 1, is the valence state ionization potentialhi) of an electron in orbital L on atom A, i.e. the ionization potential of the atom in the state characteristic of its molecular environment. In chemisorption systems, charge transfer can at times be large and must be considered. Charge transferred to (from) the adsorbate atom is taken from (placed into) metallic electron states which are not well localized. Consequently, possible large charge deviations are expected only on the adsorbate orbitals. If Pi:” is the normal occupation of the pth orbital of the valence state atom, we obtain (2.10) For the case of hydrogen, FALz - 13.6 + Pnlynni. We now consider
the off-diagonal
matrix elements
(2.11) of the Hamiltonian.
The
use of eqs. (2.7) yields
F,, g (4 -
JV2-
zA Ir
_
Ia>
RAI
+
c
Pp,Y,Ac7 +
+
c
pwY,l~
atoms A, B
atoms A, B
p~“spvYpla -
lr#v
c
P+
P
c&I
a#A
Ia>,
(2-W
A MOLECULAR
with 1~) centered
F&l = (4 -
ORBITAL
APPROACH
197
TO CHEMISORPTION
on atom A, and
ZB
9’ -
,r _
IA>+
RBj
c
c
P,Jwl
P,,Y,d +
P
atoms A, B
afomsA,B
Pp&YpoA - (4
+
9
c
(2.13)
c a#B
P#V with 12) centered on atom B. Noting that for real orbitals
(2.14)
+ F,,) 3
F,, = * P,,
again using eqs. (2.7) and assuming that x0 and xi are eigenfunctions resulting effective one electron Hamiltonian, we obtain L&[-r,+
c
(P,,-P;::mB)Yrll-I,+
c
P atomB
(P,,-P;:o~*)Y,..]s,O
P
atom A Z, Ir--R,I
(4 -4
of the
Ia>
+
:
c a=A,B
P atom A
PfipSlaYpaa+
++ c
P atom B
c P+
P,,Yh
atom A, B
PlrVSPVY/Ll~ +
+:
p,,skJ,n, c
c
01 m
Z,
Ia>.
(2.15)
c a#A,B
P#V
The last five terms contain three center integrals or the product of two center integrals and S. They are ignored in conventional EHT. The first term is explicitly included. The difference (P,, - PiFm) is, however, set equal to zero. The second term is approximated by multiplying the first by a constant K. We follow that general procedure. The difference between the atomic and molecular occupations is, however, maintained in our formulae. The conventional valuez2) of K= 1.75 - in very rough accord with viral theorem arguments - is used. The various results are not critically affected by small changes in that parameter. The binding energy of a molecular system is obtained by comparing its total energy with that of the isolated constituent atoms. A schematic HartreeFock expression for that binding energy is E,=
xEy’-(i
c Ey-) j
- (V$” - v$‘,)
+ v,, ,
(2.16)
198
ALAN
J. BENNETT,
BRUCE
MCCARROLL
AND RICHARD
P. MESSMER
where E, are the one electron energies computed approximately in eq. (2.2), V,, is the electron-electron repulsive energy, and V,, is the nuclear-nuclear repulsive energy. The seond term in eq. (2.16) corrects for the overcounting of the electron-electron interaction by the first. If the second and third term cancel, the binding energy may clearly be obtained by a simple sum over orbital energies. Goodisman42) has shown that such a cancellation approximately occurs and is a consequence of the isoelectronic principle. A slow variation of the approximate cancellation with internuclear separation would justify the neglect of those terms in calculations of relative binding energies. The binding energies reported here are determined from just the first term in (2.16). 3. Results and discussion 3.1. GRAPHITE SUBSTRATE All EHT calculations have used Slater orbitals to represent the atomic orbitals in eq. (2.1):
where parameters 2, and M, are taken from atomic data as are the ionization potentials which enter eq. (2.10). in an attempt to include solid state effects, parameters were initially obtained from a comparison with a tight-binding calculationss) of the twodimensional band structure of bulk graphite. Due to the weak coupling between hexagonal layers, it has been found that two dimensional calculations are reasonably adequate to account for many of the bulk properties of graphite with the notable exception of the conductivity, Bassani and Parraviciniss) obtained a two dimensional tight-binding band structure by fitting
I
2
Fig. 1. Sixteen carbon (16C) representation of a graphite lattice showing electron charge and reduced overlap populations (italics); underljned charge densities are those obtained with a hydrogen atom located 1 8, above position A.
A MOLECULAR
ORBITAL
a set of Sij and Hij to bulk optical
APPROACH
TO CHEMISORPTION
199
data. The use of values for the Z,, n,,
and I, derived from this parameterization lead to very diasappointing sults: In particular, the lattice is found not to be stable to deformation.
reThis
behavior emphasized that a parameterization designed to reproduce one property may be unsuitable for the proper simulation of another property42). As seen below, the customary use of the atomic parameters 22, 43) Z, = Z, = = 1.625, n = 3, I, = 19.44 eV, and IP = 10.67 eV, for the single s and three p orbitals chosen to represent the carbon atoms is more successful. Calculations were performed on various representations of the (0001) graphite substrate. The single layer 16 carbon representation of that substrate (16C) shown in fig. 1 was used in most of our work. Some results, however, were compared with those obtained using the single layer 32 carbon (32C) and stacked 32 carbon (32CS) representations shown in figs. 3 and 2 respectively. In table la, the binding energy/atom obtained using the different substrate models is listed. The agreement with the experimental value of -5 eV44) is reasonably good and the various results are similar. The 16C representation was used to calculate the binding energy/atom as a function of a, the nearest
Fig. 2. Stacked thirty two carbon (32CS) representation of a graphite lattice. Reduced overlap populations and charge distributions are the same, to three significant figures, as those shown in fig. 1.
Fig. 3.
Thirty two carbon (32C) representation of a graphite lattice showing electron charge and reduced overlap populations (italics).
200
ALAN .I. BENNETT, BRUCE MCCARROLL AND RICHARD P. MESSMER TABLE 1 (a) Representation
Binding
energy/atom
16C
4.17
32C 32CS
4.32 4.17
(eV)
(b) a (4 1.13
Binding
energy/atom
1.28
2.88 3.94
1.42
4.17
1.56
3.90 3.42
1.70
(eV)
neighbor distance. The results shown in table 1b indicate that the lattice is stable for a - 1.4 A which is the experimentally observed separation. The reduced overlap populations between various carbon atoms are shown along the lines connecting the atoms in figs 1 and 3. The reduced overlap population between interior atoms is - 1.O. A comparison with the calculated values for hydrocarbons indicates that this represents a bond order of - 1.5 between those atoms. A bond order of 1.33 is expected for the graphite structure45). Coulson 46) has considered edge rearrangement in two dimensional graphite lattices, and hypothesized that atoms such as those labeled 1 and 2 are shifted (as shown by dotted lines in fig. 1) in order to increase the lattice stability by creation of an additional bond. Our results suggest that such rearrangements are energetically unfavorable. An examination of the eigenenergies and wave functions obtained from eq. (2.2) indicates that the states may be divided into two classes. The band states extend through the substrate representation and have energies which may be correlated with the bulk band structure. The surface states47) are confined to the perimeter of the representation and have energies either within or without the range of band state energies. Their number may be roughly correlated with the number of dangling bonds on the perimeter. For large representations, these surface states would constitute a small fraction of the total number of states to be filled. They would then have a negligible effect on the Fermi level and the charge distribution in the region away from the perimeter. For the substrate representations used here, their effect is large. The range and distribution of band eigenvalues calculated differ from those
A MOLECULAR
given by Bassani adjustments structure.
ORBITAL
and Parravicinisa).
permitted,
precludes
APPROACH
201
TO CHEMISORPTION
Our use of atomic parameters, an accurate
reproduction
with no
of their
band
The total pi-electron
band width composed of p orbitals directed out of the plane is found to be N 11.61 eV (12.3 eV for 32C representation) N 17.8 eV (17.0 eV for the 32C representation) and the total filled bandwidth as compared to Bassani and Parravicini’s values of 13.6 eV and 14.3 eV respectively. The highest filled energy level occurs at - 10.0 eV (- 10.4 eV for the 32C representation) as opposed to the experimental value of N -4.8 eV48). This rather severe lack of agreement is not surprising, since many body correlation effects are not explicitly included in our calculations. For the small representations used here two other effects change the work function. When obvious surface (i.e. perimeter) states are ignored in filling the energy levels, the highest occupied level is moved toward the experimental value. These surface states are responsible for the enhanced electron charge on perimeter atoms shown in figs. 1 and 3. Also the finite spacing of the energy levels in a small representation affects the calculated work function. If the finite representation is assumed to be in electrical contact with the remainder of the substrate, thermodynamics demands that its Fermi level and that of the substrate be equal. Consequently electrons flow from the bulk into the surface region to create a charge excess which causes a Hartree potential. That potential self-consistently adjusts the representation’s energy levels in order to control the charge flow. Simple Poisson equation considerations indicate that a small excess of charge (6 1 cl/atom) is sufficient to shift the energy levels on the order of electron volts. 3.2. HYDROGEN CHEMISORPTION The diagonal matrix element associated with the hydrogen s orbital is given by eq. (2.4). The magnitude of yiil can be calculated using hydrogenic orbitals and found to be 8.5 eV. As suggested by Newns, we use ylnn = + (I” -AH) = 6.45 eV, a somewhat better measure of the physically desired quantity since it takes some account of correlation effects. For hydrogen positions near a metal surface, even greater reductions in yiln may occur49). We consider the interaction of the adsorbate with the substrate representation in electrical contact with the bulk, i.e. with a plateau region on an otherwise perfect surface. We demand that the highest filled energy level, EFs, in the substrate representation be close to the correct bulk Fermi level EFB = - 4.8 eV. The simplest means of assuring this equality is a shift of each representation level by (E,, -I$,). A severe and probably artificial reduction in the cohesive energy of the substrate representation would then be obtained from eq. (2.2) et seq. by use of the modified carbon levels.
202
ALAN
.I. BENNETT,
BRUCE
MCCARROLL
AND
RICHARD
The physically correct difference between adsorbate crucial for investigating the ionicity of their interaction,
I’. MESSMER
and substrate levels, can alternatively be
maintained by changing the effective adsorbate levels by - (E,, -I&). This approach is used here. There are resultant changes in the magnitude of various calculated binding energies, but the relative binding energies which are the main output of our calculations remain approximately invariant. For investigation of perfect surfaces, an approach, currently under investigation, is to assume periodic boundary conditions in the plane, and thus eliminate surface states. The self-consistent charge and binding energies were found after two or three iterations. The hydrogen binding energies as a function of the distance Z above various points in the lattice plane are shown in figs. 4a, 4b, and 4c for the point A (at a carbon atom), point B (between two nearest neighbor carbon atoms), and point C (in the middle of a hexagonal cell). The points A, B, and C are indicated in fig. 1. The dashed lines in fig. 4 indicate the charge on the hydrogen adsorbate. The calculations were performed on the 16C representation but some results for the 32C representation are shown by crosses in the figure. (0)
‘H -c
5
uY
__---
0.5
1.0
;i
I% E B 1.5 g
-I
-I t
ZCil Fig. 4. The solid lines show binding energy versus distance from the graphite surface for atomic hydrogen located above points (a) A, (b) B, and (c) C on the 16C representation. The dashed lines in the figure indicate the net hydrogen charge. The crosses show the results of calculations on the 32C representation.
A MOLECULAR
The
converged
effective
ORBITAL
APPROACH
hydrogen
203
TO CHEMISORPTION
ionization
potentials
are rather
close
to the atomic value of 13.6 eV. The conventional, and generally successful, EHT use of that value would thus not strongly affect our results. The calculation at the point A is clearly inadequate for very small values of 2 when the adsorbate and carbon atom coincide. At the other extreme, molecular orbital calculations without configuration interaction fail to give the correct dissociation behavior at large separations. The underlined numbers in fig. 1 show the charge distribution on the 16C lattice when a hydrogen atom is located 1 A above the point A. Comparison with the other charges shown in that figure indicates that little charge redistribution occurs due to the presence of the hydrogen. This justifies the self-consistent adjustment of only the hydrogen atomic energy level. Fig. 4 and similar curves for other points indicates that the maximum binding always occurs for Z-1 A. An equi-binding energy surface obtained using the 16C representation is shown in fig. 5. We again emphasize that differences between the binding energies shown in the figure are of most significance. The maximum binding occurs above a carbon atom. The barrier for hydrogen motion along the lines connecting nearest neighbor carbons is low, and hydrogen should avoid the center of the hexagonal cell in preference for positions above the carbon atoms. The use of larger representations is found to affect some details but not the essential features of this figure.
C
C
c 1.121.09
I.14 Fig. 5.
1.05
1.09 I.12
c I.14
Equi-binding energy contours (eV) for atomic hydrogen adsorbed on the (16C) graphite substrate. Positive energies correspond to binding.
204
ALAN
This prediction
J. BENNETT,
contrasts
BRUCE
MCCARROLL
AND RICHARD
sharply with conventional
P. MESSMER
hard sphere reasoning,
which maintains that adsorbed species tend to assume a surface configuration that maximizes the number of nearest substrate atoms. Our result suggests that orbital geometry is a crucial ingredient in the determination of surface binding sites. 3.3. OTHERADSORBATES Calculations were performed for nitrogen and oxygen adsorbates similar to those performed for hydrogen adsorbates. However, the presence of such electrophilic species has a profound effect on the charge distribution of the substrate representation. Self-consistent adjustment of only the adsorbate matrix elements was found inadequate to obtain convergence. Self-consistent adjustment of substrate atom matrix elements is apparently required. The results of preliminary calculations do indicate that electrophilic adsorbate atoms, in contrast to hydrogen, are not repelled at the center of the hexagonal graphite cell. Preliminary calculations were also carried out for other species. For example, the results predict that CO is not adsorbed on top of the substrate but seems to bind only at edge atoms with the carbon atom of the CO molecule bonding to a carbon atom of the substrate. LEED results 50) suggest that CO does not bind on top, and other experimental data indicate that edgebound CO may be a participating species in the oxidation of carbon51). Despite this encouraging comparison between theory and experiment, it must be remembered that no self-consistent adjustments were attempted in these calculations, so that the conclusions for the CO-graphite system must be taken as tentative. 3.4. METHANE
FORMATION
The results of experimental studies on the interaction of hydrogen atoms with graphite and carbon powders indicate that the main reaction product is gaseous methane CH,s*), which is in accord with thermodynamic expectations. A quantum mechanical examination of a reaction path should predict intermediate steps between the starting materials and the final reaction products. At each step, the geometric configuration of maximum stability should be apparent, and any energetic barriers to progress through the sequence of steps should emerge from the calculations. In principle, all viable alternate reaction paths should be calculated. Of course, the energetics of the terminal configurations (products) must be consistent with thermodynamic predictions. In practice, the possible geometric alternatives of a given step of even our
A MOLECULAR
simple system is enormous,
ORBITAL
APPROACH
TO CHEMISORPTION
and the detailed investigation
205
of all configurations
is clearly impractical. Therefore, the reaction path outlined below for the formation of CH, from hydrogen and a graphite crystal must be considered illustrative rather than definitive. The configurations considered are those of lowest energy picked from a number of reasonable possible alternatives. A single hydrogen atom binds with an energy of about 4 eV to edge carbon atoms, i.e. much more strongly than to atoms in the center of the representation. This behavior suggests that the strongest interactions occur at the edges and no doubt other such portions of a real crystal surface where so-called free valences are available. Indeed, reactions of atomic hydrogen with crystalline graphite surfaces produce surface pits whose size increases with increasing exposure to hydrogen atomss2). The experimental behavior thus also suggests that the abstraction of carbon atoms from the surface occurs initially at surface imperfections, and then continues at edges thus formed. The stabilities shown in fig. 6 are determined by subtracting from the calculated energy of a given configuration the sum of the total energy for a bare 16 carbon lattice and - 1.36 n where II is the number of bound hydrogens. The 16C lattice and n-gas phase hydrogens is therefore the zero of energy for the comparisons. Only rough charge consistency was obtained.
*
METHANE
Fig. 6.
Simulated reaction path for the formation of methane from a 16C lattice and 8 hydrogen atoms; see text.
206
ALAN
J. BENNETT,
BRUCE
MCCARROLL
AND RICHARD
P. MESSMER
The addition of the first hydrogen is most stable on carbon D (fig. 6). It lies in the plane of the sheet. The second hydrogen binding at E provides the most stability and the addition of the third hydrogen provides the planar arrangement shown in the figure. The addition of more hydrogens with the aim of incorporating carbon D into a methane molecule proceeds as shown in the figure. Any carbon with two hydrogens has them equidistant above and below the plane of the sheet with an H-C-H angle of 120 O.The three hydrogens on carbon D are situated at three corners of a tetrahedron with carbon D as the center. The most stable aspect is with one of the three hydrogens in the plane of the sheet. The fourth hydrogen is added to carbon D from above the plane. Several classes of configurations were not examined. In particular, as hydrogens are added to the system, it is reasonable to expect the C-C bond length to increase to approximately 1.54 A, which would parallel behavior observed for organic molecules of increasing hydrogenation. However, other than moving carbon D away from the lattice (fig. 6) in the plane, no attempt is made to adjust C-C bond lengths as the hydrogen adsorption progresses. Out-of-plane distortions in carbon rings that may occur upon increasing hydrogenation are also ignored. The events depicted certainly constitute a reaction path. But they undoubtedly do not constitute the most probable reaction path, except fortuitously. However, the extended Hiickel calculations do enable us to energetically distinguish between configurations for a fixed number of added hydrogens and exhibit at least one energetically favorable means of methane production. Further, the results of calculations for terminal configurations involving CH, or C,H, as the products reflect both the thermodynamic predictions and the experimental results of methane as the most stable product. 4. Summary An extended Hiickel theory has been used to treat chemisorption. The present work is characterized by the self-consistent adjustment of only adsorbate atom energy levels and the use of a finite representation of the substrate. The former approximation seems adequate for treating atomic hydrogen on graphite. However, electrophilic adsorbates require a more complete self-consistent treatment in which all energy levels are adjusted. The finite representation of the substrate does allow semiquantitative investigations of adsorption on both perfect and imperfect surfaces. For small representations the population of electron states localized near the perimeter can significantly perturb the distribution of electrons from that of a larger representation. We emphasize that absolute values of the adsorbate binding
A MOLECULAR
ORBITAL
APPROACH
TO CHEMISORPTION
207
energies are only crudely determined. The calculated differences between the energies of various configurations are, however, relatively invariant to various details of the procedure. This is of significance because many useful predictions can be made on the basis of relative binding energies. For example, a feature that clearly emerges from the results of our calculations is that the use of lattice geometry alone is insufficient to predict the binding site on a given surface for a given adsorbate. The directional characteristics of the electronic interactions are indeed controlling, so that, for example, a position on the surface that maximizes the number of nearest neighbors is not always a position of minimum energy. Although fraught with technical problems the simple EHT method yields results which suggest that with suitable modifications of the method we may eventually hope for detailed predictions of various adsorbate properties.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11)
12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25)
See, for example, T. A. DeIchar and G. Ehrlich, J. Chem. Phys. 42 (1965) 2686. E. W. Plummer and T. N. Rhodin, J. Chem. Phys. 49 (1968) 3479. G. Ehrhch and F. Hudda, J. Chem. Phys. 44 (1966) 1039. H. D. Hagstrum and G. E. Becker, Phys. Rev. Letters 22 (1969) 1054. G. C. Bond, Cutulyis &y Merals (Academic Press, New York, 1962). E.g. L. D. Schmidt and R. Gomer, J. Chem. Phys. 45 (1966) 1605. E.g. H. E. Neustadter and R. J. Bacigalupi, Surface Sci. 6 (1967) 246. G. Ehrlich, H. Heyne and C. F. Kirk, in: The Sfructare and Chemistry ofSolid Sarfaces, Ed. G. A. Somorjai (Wiley, New York, 1969) p. 49-l. S. Glasstone, K. J. Laidler and H. Eyring, The Theory of Rate Processes (McGraw Hill, New York, 1941). J. Horiuti and K. Miiller, J. Res. Inst. Catalysis, Hokkaido Univ. 16(1968) 605. R. Gomer and J. R. Schrieffer (private communication) have recently formulated a many electron Heitler-London approach which is being applied to the chemisorption problem. T. Toya, J. Res. Inst. Catalysis, Hokkaido Univ. 8 (1960) 209; 6 (1958) 308. K. F. Wojeiechowski, Acta Phys. Polonica 29 (1966) i19; 33 (1968) 363. D. M. Newns, Phys. Rev. 178 (1969) 1123. T. B. Grimley, Proc. Phys. Sot. (London) 90 (1967) 751; 92 (1967) 776. D. A. Dowden, in: Coloquio Sobre Quimica Fisica de Procesos en Superficies Solidas (Libreria Cientiflca Medinaceli, Madrid, 1965). K. Jug, Theoret. Chim. Acta. (Berlin) 14 (1969) 91. H. H. Jaffe, Accounts of Chemical Research 2 (1969) 136. A. J. Bennett and L. M. Falicov, Phys. Rev. 151 (1966) 512. J. W. Gadzuk, in: The Structure and C~e~jstry of Solid Surfaces, Ed. G. A. Somorjai (Wiley, New York, 1969) p. 43-l. P. 0. Lowdin, Advan. Phys. 5 (1956) 1. R. Hoffmann, J. Chem. Phys. 36 (1962) 2179; 36 (1962) 2189; 40 (1963) 2474. T. B. Grimley, in: Molecular Processes on Solid Surfaces, Eds. E. Drauglis, R. D. Gretz and R. J. Jaffee (McGraw-Hill, New York, 1969) p. 299. P. Markov and D. Lazarov, Compt. Rend. Acad. Butgare Sci. 22 (1969) 455; D. Shopov, A. Andreev and D. Petkov, J. Catalysis 13 (1969) 123. A. Rosenthal and R. Hoffmann (private communication).
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J. BENNETT,
BRUCE
MCCARROLL
AND RICHARD
P. MESSMER
26) B. J. Duke, Theoret. Chim. Acta (Berlin) 9 (1968) 260. 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 3% 39) 40)
41) 42) 43) 44) 45) 46) 47) 48) 49) 50) 51) 52)
A. J. Bennett, B. McCarroll and R. P. Messmer, to be published. G. Blyholder and C. A. Coulson, Trans. Faraday Sot. 63 (1967) 1782. B. R. Cooper and A. J. Bennett, Phys. Rev., to be published. J. C. Slonczewski and P. R. Weiss, Phys. Rev. 109 (1958) 272. J. W. McClure, Phys. Rev. 108 (1957) 612. F. Bassani and G. Parravicini, Nuovo Cimento 50B (1967) 95. F. J. Vastola, P. L. Walker and J. P. Wightman, Carbon 1 (1963) 11. P. S. Gill, R. E. Toomey and H. C. Moser, Carbon 5 (1967) 43. E. B. Moore Jr. and C. M. Carlson, in: Lattice Defects and Their Inleructions, Ed. R. R. Hasiguti (Gordon and Breach, New York, 1967) p. 167. G. Blyholder and C. A. Coulson, Theoret. Chim. Acta (Berlin) 10 (1968) 316. M. D. Newton and W. N. Lipscomb, J. Am. Chem. Sot. 88 (1966) 2353, 2361, 2367. C. C. J. Roothaan, Rev. Mod. Phys. 23 (1951) 69. R. S. Mulliken, J. Chem. Phys. 23 (1955) 1833. R. S. Mulliken, J. Chim. Phys. 46 (1949) 497, 675.; J. W. Richardson and R. E. Rundle, Report ISC-830, Ames Lab., Iowa State College USAEC, Technical Information Extension, Oak Ridge, Tenn. 1956. R. S. Mulliken, J. Chem. Phys. 2 (1934) 782. J. Goodisman, J. Am. Chem. Sot. 91 (1969) 6552. J. A. Pople and G. A. Segal, J. Chem. Phys. 43 (1965) 5136. M. A. Kanter, Phys. Rev. 107 (1957) 655. L. Pauling, The Nature ofthe Chemical Bond (Cornell University Press, Ithaca, 1960). C. A. Coulson, in: Proc. Fourth Conf. on Carbon (Pergamon Press, New York, 1960) p. 215. J. Koutecky and M. Tomasek, Phys. Rev. 120 (1960) 1212. V. S. Fomenko, Handbook of Thermionic Properties (Plenum Press Data Division, New York, 1966). A. Heeger, in: Solid Rate Physics, Vol. 23 (Academic Press, New York, 1969). J. J. Lander and J. Morrison, Surface Sci. 6 (1967) 1. See, for example, L. Schmidt, H. P. Boehm and U. Hoffman, in: Proc. Third Conf. on Carbon (Pergamon Press, New York, 1959) p. 235. B. McCarroll and D. W. McKee, Nature 225 (1970) 723; Carbon, to be published.